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Points :: points

points -- produces the ideal and initial ideal from the coordinates of a finite set of points

Synopsis

Description

This function uses the Buchberger-Moeller algorithm to compute a grobner basis for the ideal of a finite number of points in affine space. Here is a simple example.
i1 : M = random(ZZ^3, ZZ^5)

o1 = | 1 3 4 3 3 |
     | 1 8 5 6 2 |
     | 5 1 0 0 8 |

              3        5
o1 : Matrix ZZ  <--- ZZ
i2 : R = QQ[x,y,z]

o2 = R

o2 : PolynomialRing
i3 : (Q,inG,G) = points(M,R)

                    2                     2        2   3           29 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  -
                                                                  138    
     ------------------------------------------------------------------------
     125    125    211    375        25 2   70    70    14    210   2   103 2
     ---x - ---y - ---z + ---, x*z - --z  - --x - --y - --z + ---, y  + ---z 
      69     69     46     23        69     69    69    23     23       138  
     ------------------------------------------------------------------------
       20    739    337    630        20 2   289    151    44    177   2  
     + --x - ---y - ---z + ---, x*y + --z  - ---x - ---y - --z + ---, x  +
       69     69     46     23        69      69     69    23     23      
     ------------------------------------------------------------------------
      5 2   147    14    33    150   3   257 2   140    140    514    1260
     --z  - ---x + --y - --z + ---, z  - ---z  - ---x - ---y + ---z + ----})
     23      23    23    23     23        23      23     23     23     23

o3 : Sequence
i4 : monomialIdeal G == inG

o4 = true

Next a larger example that shows that the Buchberger-Moeller algorithm in points may be faster than the alternative method using the intersection of the ideals for each point.

i5 : R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]

o5 = R

o5 : PolynomialRing
i6 : M = random(ZZ^5, ZZ^150)

o6 = | 4 5 9 8 0 5 6 0 1 4 3 0 4 9 3 4 1 2 3 7 4 3 0 0 8 5 5 2 5 9 1 3 4 8 0
     | 2 0 4 4 7 6 0 9 7 0 1 1 9 7 5 0 2 2 1 8 2 8 1 6 5 1 1 9 2 3 4 8 6 4 8
     | 7 4 7 9 9 3 7 6 4 1 9 9 6 1 8 3 8 2 6 8 1 9 4 7 8 6 2 1 9 7 5 1 7 3 7
     | 8 6 4 1 2 6 9 0 9 9 5 7 6 0 1 0 2 3 3 7 3 8 9 9 9 6 0 6 5 2 5 8 6 5 5
     | 0 2 3 5 2 2 7 9 5 4 3 1 6 2 9 3 0 6 4 7 5 0 2 5 8 0 4 2 0 4 3 4 3 9 0
     ------------------------------------------------------------------------
     2 9 3 9 3 8 6 7 2 6 0 3 9 2 3 5 0 8 3 5 2 9 2 3 0 6 3 3 8 9 7 0 3 6 1 2
     7 3 7 1 1 9 1 4 4 1 0 1 4 4 9 9 0 2 3 4 5 2 7 0 1 6 8 2 8 7 3 8 6 6 2 8
     3 3 4 0 0 8 5 8 8 6 7 4 1 3 9 3 4 1 1 2 4 8 8 9 2 3 4 3 4 5 5 4 6 9 5 7
     9 6 1 7 5 9 3 9 0 7 8 8 1 1 6 2 1 4 1 1 7 8 2 1 6 6 6 5 9 0 0 4 3 4 9 3
     0 4 0 1 9 4 4 9 6 3 4 2 4 3 4 9 6 6 0 0 4 6 8 8 1 4 3 2 1 5 5 6 4 6 4 4
     ------------------------------------------------------------------------
     1 3 2 3 1 8 1 2 8 5 0 1 9 8 8 8 1 7 1 1 8 2 0 0 3 7 3 7 3 2 2 1 0 9 1 1
     2 9 6 3 6 9 2 5 7 7 7 5 7 8 1 9 9 6 1 3 3 9 6 8 9 0 4 2 3 0 9 2 0 0 7 2
     7 7 5 4 1 3 3 0 0 7 0 1 0 2 6 3 2 1 4 4 6 3 4 9 9 8 0 6 4 1 3 9 2 0 3 3
     7 6 5 3 1 9 5 7 7 6 8 4 7 6 6 0 8 3 9 7 3 8 3 2 6 1 9 6 0 8 1 5 9 7 6 3
     8 6 0 9 0 4 2 7 0 0 9 9 9 8 6 5 1 2 1 9 8 3 5 3 5 0 7 5 9 5 6 7 3 2 8 1
     ------------------------------------------------------------------------
     0 5 0 2 7 2 2 4 1 8 2 7 7 9 8 1 7 2 2 7 2 7 2 5 5 6 5 5 3 4 6 9 7 5 7 8
     3 4 2 4 1 1 6 2 3 4 8 0 6 6 2 1 8 5 1 9 7 1 0 3 7 5 9 9 1 0 4 9 1 9 4 3
     3 8 1 2 7 0 7 7 0 5 6 8 4 9 5 9 2 3 3 7 2 2 0 4 4 7 4 8 6 4 8 6 6 9 3 3
     6 5 6 8 6 6 2 9 0 9 4 9 4 6 2 1 3 4 9 7 2 2 3 2 3 7 5 2 8 7 4 6 6 3 2 7
     1 8 7 6 5 6 6 3 2 5 6 6 3 8 7 6 3 3 1 9 8 3 9 5 2 5 0 5 7 1 1 1 0 3 3 6
     ------------------------------------------------------------------------
     0 9 6 1 9 9 8 |
     7 4 1 1 0 2 1 |
     5 9 3 1 0 7 0 |
     0 8 8 4 2 1 7 |
     3 0 9 7 0 0 6 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 22.0441 seconds
i8 : time C = points(M,R);
     -- used 1.88905 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :